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| Mirrors > Home > MPE Home > Th. List > ovresd | Structured version Visualization version GIF version | ||
| Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| ovresd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
| ovresd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
| Ref | Expression |
|---|---|
| ovresd | ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ovresd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
| 2 | ovresd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
| 3 | ovres 7599 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 × cxp 5683 ↾ cres 5687 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: sscres 17867 fullsubc 17895 fullresc 17896 funcres2c 17948 rngchom 20623 ringchom 20652 rhmsubclem4 20688 irinitoringc 21490 psmetres2 24324 xmetres2 24371 prdsdsf 24377 xpsdsval 24391 xmssym 24475 xmstri2 24476 mstri2 24477 xmstri 24478 mstri 24479 xmstri3 24480 mstri3 24481 msrtri 24482 tmsxpsval 24551 ngptgp 24649 nlmvscn 24708 nrginvrcn 24713 nghmcn 24766 cnmpt1ds 24864 cnmpt2ds 24865 ipcn 25280 caussi 25331 causs 25332 minveclem2 25460 minveclem3b 25462 minveclem3 25463 minveclem4 25466 minveclem6 25468 ftc1lem6 26082 ulmdvlem1 26443 abelth 26485 cxpcn3 26791 rlimcnp 27008 hhssnv 31283 madjusmdetlem3 33828 qqhcn 33992 qqhucn 33993 ftc1cnnc 37699 ismtyres 37815 isdrngo2 37965 naddcnffo 43377 |
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